A min-time analysis of three trajectories with curvature and nonholonomic constraints using a parallel parking criterion

This paper presents a comparison of three s-shaped curves for parallel parking a car subject to nonholonomic constraints. The comparison is based on the rate at which the car approaches the curb (the parking criterion). The curves (two circular arcs, a cosine curve and a quintic) are shown to accommodate an initial and final position, but only the quintic accommodates initial and final curvature. The time penalty in straightening out the wheels at the maneuver end-points (or midpoint) causes the other curves to be suboptimal. Representative examples of the curves are shown. Also illustrated is the effect of the turning radius of the car on the time required to park it.